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Numerical Energy Dissipation for Time-Fractional Phase-Field Equations

Numerical Energy Dissipation for Time-Fractional Phase-Field Equations

Year:    2025

Author:    Chaoyu Quan, Tao Tang, Jiang Yang

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 3 : pp. 515–539

Abstract

The numerical integration of phase-field equations is a delicate task which needs to recover at the discrete level intrinsic properties of the solution such as energy dissipation and maximum principle. Although the theory of energy dissipation for classical phase field models is well established, the corresponding theory for time-fractional phase-field models is still incomplete. In this article, we study certain nonlocal-in-time energies using the first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete fractional energy law and a discrete weighted energy law. The extension for a $(2−α)$-order L1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate that the energy bound is preserved for the L1 schemes with nonuniform time steps. Several numerical experiments are carried to verify our theoretical analysis.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.2311-m2021-0199

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 3 : pp. 515–539

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    25

Keywords:    Time-fractional phased-field equation Allen-Cahn equations Cahn-Hilliard equations Caputo fractional derivative Energy dissipation.

Author Details

Chaoyu Quan

Tao Tang

Jiang Yang